Midterm 1 Practice Problems

Transcription

1 Midterm 1 Practice Problems 1. Calculate the present value of each cashflow using a discount rate of 7%. Which do you most prefer most? Show and explain all supporting calculations! Cashflow A: receive $60 today and then receive $60 in four years. Cashflow B: receive $12 every year, forever, starting today. Cashflow C: pay $50 every year for five years, with the first payment being next year, and then subsequently receive $30 every year for 20 years. Cashflow D: receive $9 every other year, forever, with the first payment being next year. 2. Consider a project costing $1m each year from year 1 to year T. Then starting in year T+1, the project will generate a profit of $700k each year, forever. a) Write a formula for the present value of this project with a discount rate of r. b) Write a formula in terms of r for the value of T at which you break even (ignoring the issue of whether T is an integer). 3. Suppose you had $10,000 to invest for one year. You are deciding between a savings account with a 2% annual interest rate compounded daily (alternative A) and one with a 2% annual interest rate compounded monthly (alternative B). You are about to invest in the alternative A, but then you realize that since that bank is in downtown Chicago, you ll need to spend an extra $1 for parking when opening the account. Alternative B does not have this cost (it s a bank in Evanston). Should you change your decision or stick with alternative A? Show and explain all supporting calculations! 4. What is the effective annual interest rate in each situation? a. A savings account with 4% annual interest rate compounded daily (assume a year consists of 365 days)? b. A savings account with 4% annual interest rate compounded monthly? 5. Consider the following cashflow stream and a bank account paying 3% annual interest. What is the present value? Is the account value ever negative? Year Cashflow Which of the following cashflows do you most prefer using a discount rate of 10%? Using a discount rate of 1%? Show and explain all supporting calculations! Cashflow A: receive $10 every year, forever, with the first payment next year Cashflow B: receive $19 every other year, forever, with the first payment being next year

2 Cashflow C: pay $5 every year for 20 years, with the first payment being today, and then subsequently receive $30 every year for 20 years. Cashflow D: receive $70 today and then receive $50 in five years. 7. Irene Engels recently graduated with an MBA. In August 2007, she borrowed $50,000, and she borrowed another $50,000 in August Her student loan has an annual interest rate of 2% compounded monthly. Irene doesn t make any payments on her student debt until she starts a lucrative Wall St. job. Then starting in September 2009 she makes a payment of $1000 every month. Now bonus time is coming near. For January 2010 she plans to make another $1000 payment (her 5 th ) and also apply her bonus to the debt. How big must her bonus be so that she will have completely paid-off the debt at the end of this January? 8. You are analyzing the value of the company Twitter using a 15% discount rate. You expect its cashflows over the next 4 years to be as shown below and you estimate its NPV as $1B. Explain. Year Cashflow 0-20M 1-10M M 4 40M 9. A bank offers a savings account with a 3% annual interest rate, compounded monthly. 9.1 What is the effective annual interest rate? 9.2 Stu wants to open a savings account and make one deposit now that will enable him to withdraw $700 to go on vacation 5 months from now and $2000 for a deposit on a rental apartment when he starts working in 20 months from now. How much money does Stu need to deposit now? 10. If the discount rate is 12%, what is the present value of receiving $1000 per year at the end of each of the next 8 years? 11. Using a discount rate of 5%, what is the net present value of the following cashflow stream? Year Cashflow You bought a $200k condo. You got a 15-year fixed-rate mortgage and made a 20% down payment.

3 a) What is your monthly payment? b) Would the monthly payment be bigger or smaller with a 30-year mortgage at the same interest rate? 13. Consider the following cashflow stream and a bank account paying 10% annual interest. Today the account has $9. Year Cashflow What is the largest amount ever in the account? 14. Calculate the PV of the following cashflows using a 7% discount rate. a) 30 payments of 100 starting 5 years from today b) you pay 10/yr for 3 years with the first payment being today, and then starting a year from today you will receive $6/yr for 6 years. 15. Suppose that a construction project costs $10m (in present value) if you start it today. What are the savings (in present value) of delaying it by 3 years. Assume a 10% discount rate and that the price remains the same. 16. A 1% monthly rate of return is equivalent to what annual rate (compounded yearly)? 17. You ve taken out a 30-year mortgage for 120k with a 4.2% rate. a) What s your minimum monthly payment? b) You ve paid $700/mo for one and a half years. You re now trying to refinance. What s the principal remaining on your mortgage? 18. You and two friends are considering buying a house in Chicagoland to live here together after you graduate. You can get a 15-year fixed-rate mortgage with a mortgage rate of 5% if you make a 20% down payment on the house. You will split the monthly mortgage payment equally among the three of you. Each of the three of you can afford to contribute up to $1,000 per month towards the mortgage payment. You each have $10,000 available towards the down payment. How expensive a house can you afford to buy? 19. For all the parts to this problem, let the annual discount rate be 5%. a) Find the present value of the following cashflow: receive $10 every year for 30 years with the first payment being 10 years from now. b) Find the present value of the following cashflow: receive $10m now and the same amount a year from today and pay $3m a year forever with the first payment being a year from today. c) Consider the following two cashflows. For cashflow A, you receive $10 every year for 5 years with the first payment being today. For cashflow B, you receive x

4 dollars every year forever with the first payment being today. What is the value of x in order for cashflow B to have the same present value as cashflow A? 20. Today, you re in charge of the nation s finances. Suppose that projected 2015 shortfall is $418 billion and projected 2030 shortfall is $1,345 billion. In present value terms, how large is the difference of the two budget shortfalls? Assume a 3% discount rate. 21. Suppose that you borrowed $20k for 36 months to buy a car last year at an annual interest rate of 5% compounded monthly. a) What is the amount of monthly payment? b) Calculate the effective annual interest rate for both the car loan and for a rate of 6% compounded quarterly. Which is larger? c) You made monthly payments for the last 12 months. But you still have to make 24 more payments. What is the present value of the remaining payments? 22. Suppose that you consider some mortgage options. The price of home is $200k. Calculate your monthly payments for each option: - Option A: 20% down payment at 15-year fixed annual rate of 4% - Option B: 15% down payment at 30-year fixed annual rate of 4.5% - Option C: 10% down payment at 30-year fixed annual rate of 6% 23. Suppose that an account has $6m now. The money is invested and obtains a return of 2%. Your business projections are that in year one you take out $2m, in year two you take out $0.7m, in year three you add $1m to the account, and in year four you add $4m to the account. Calculate the amount of money in the account a year from now, two years from now, three years from now, and four years from now. 24. Consider a 30-year mortgage with a 5% interest rate and a 20% down payment. If you can afford a $1000 monthly payment, how expensive a house can you buy? 25. Suppose you decide to buy a $200,000 condo. You make a 10% down payment and take out a 30-year fixed-rate mortgage at 6%. a) What is your monthly payment? b) Suppose in three years the rate for a 15-year fixed rate mortgage is 5%. What would your new monthly payment be if you decided to refinance then? 26. For all the parts to this problem, let the annual discount rate be 3%. a) Find the present value of the following cashflow: receive $13,240 every year for 20 years with the first payment being 45 years from now. b) Consider the following cashflow: receive x dollars now and the same amount in a year from today, and pay $300k a year forever with the first payment being a year from today. What is the value of x in order for the present value of the cash flow to be 0?

5 27. Consider a stimulus program that intends to spend $300 billion every year, for three years. Assuming a 3% discount rate, a) what is the present value of the program? b) how much would the present value increase if the $300 billion were spent at the beginning of each year rather than at the end? 28. You are running a small business. At the beginning of the month you have $1000. At the end of the first week you have revenues of $2200 and expenses of $1000 for that week. In the second week your revenues are $2000 and your expenses are $700. In the third week your revenues are $2100 and your expenses are $1100. In the fourth week, your revenues are $2200 and your expenses are $3000 (they are higher as you need to pay the rent). You have a checking account earning 1% annually compounded weekly. a) How much money do you have at the end of the four weeks? b) What is the minimum balance of the account over those four weeks? Does it ever drop below $1000? 29. What is the effective annual interest rate of a) a car loan with a 5% annual interest rate compounded monthly? b) a credit card with a 24.7% annual interest rate compounded monthly? 30. The price of home is $250k. Calculate your monthly payment if you get a mortgage with a 20% down payment at 15-year fixed annual rate of 3%. 31. Five years ago you bought a home and took out a 30 year mortgage for 150k at 6%. Suppose you ve made monthly payments of $1200. (This may be higher than the minimum monthly payment.) a) What is the remaining principal today? b) What is the monthly payment if you would refinance the mortgage with a new 15-year mortgage at 3%? 32. Consider a 30-year mortgage with a 5% interest rate and a 20% down payment. If you can afford a $1000 monthly payment, how expensive a house can you buy? How large is the down payment? Solutions 1. PV of A = 60+60*1.07^-4 = $ PV of B = 12+12/0.07= $ PV of C = -50/0.07*(1-1.07^-5)+30/0.07*(1-1.07^-20)*1.07^-5 = $21.59

6 PV of D = 9/(1.07^2-1)*1.07 = $66.46 The PV of cashflow B is largest and thus most preferred. 2. PV=$-1m*(1-(1+r) -T )/r + $700k*(1+r) -T /r PV=0 implies ($700k+$1m)(1+r) -T = $1m so T = log 1.7 / log (1+r) 3. FV of alternative A: $9999*(1+0.02/365)^365=$10, FV of alternative B: 10,000*(1+0.02/12)^12=$10, Since the FV of B is greater than the FV of A, you should change your decision and go with alternative B. 4a. (1+0.04/365)^365-1= = 4.08% 4b. (1+0.04/12)^12-1= = 4.07% 5. Present value equals 8+2*1.03^-1 + 4*1.03^-2 15*1.03^ *1.03^-4 = If the account value is ever negative, then it will be at the end of year 3. The present value up cashflows through year 3 is 8+2*1.03^-1 + 4*1.03^-2 15*1.03^-3= Since this is negative, the account will be negative at the end of year The present value of cashflow A is 10/r, or 100 when r= 10% and 1000 when r=1%. The two period interest rate is s=(1+r)^2-1, or 21% when r=10% and 2.01% when r=1%. The present value of cashflow B is (1+r)*19/s where the 1+r factor accounts for the fact that the first payment is in one year (half of a two year period). Thus the present value is when r=10% and 955 when r=1%. The present value of cashflow C is -5-5/r*(1- (1+r)^-19)+(1+r)^-19*(30/r*(1-(1+r)^-20), or when r=10% and 357 when r=1%. The present value of cashflow D is 70+50*(1+r)^-5, or 101 when r=10% and 118 when r=1%. Thus when r=10% then cashflow D is preferred and when r=1% then cashflow A is preferred. 7. Let r=0.02/12 be the monthly interest rate. The future value of the debt at the end of August 2009 is 50000*(1+r)^ *(1+r)^12 = 103,048. The present value at the end of August 2009 of the future payments is 1000/r*(1-(1+r)^-5) = Thus the value of the debt at the end of August 2009 is 103, =98,073. Thus the future value of the debt at the end of January 2010 is 98,073*(1+r)^5=$98,893. A bonus this big would allow her to pay off the debt. 8. Clearly the present value of the cashflows over the next 4 years is less than $1B. So to have a present value of $1B the cashflows after year 4 must be pretty big. Another way of saying the same thing is that the value of Twitter, X, at the end of year 4 must be quite high. We can actually calculate X. The future value X at year 4 is X=(1B-20M)*1.15^4 10M*1.15^3+12M*1.15^1+40M = 1.753B. 9.1 Answer: Since the annual interest rate a = 3%, compounded in m =12 periods, then the effective annual interest rate i is

Chapter 2 - Sample Problems 1. If you wish to accumulate $140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%? 2. What will $247,000 grow to be in

Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values

The Time Value of Money Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original

6-1 Chapter 6 Time Value of Money Concepts 6-2 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in

Chapter 5 DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS The basic PV and FV techniques can be extended to handle any number of cash flows. PV with multiple cash flows: Suppose you need $500 one

Problem Set: Annuities and Perpetuities (Solutions Below) 1. If you plan to save $300 annually for 10 years and the discount rate is 15%, what is the future value? 2. If you want to buy a boat in 6 years

How to Calculate Present Values Michael Frantz, 2010-09-22 Present Value What is the Present Value The Present Value is the value today of tomorrow s cash flows. It is based on the fact that a Euro tomorrow

MGT201 Lecture No. 07 Learning Objectives: After going through this lecture, you would be able to have an understanding of the following concepts. Discounted Cash Flows (DCF Analysis) Annuities Perpetuity

International Financial Strategies 1 Future Value and Compounding Future value = cash value of the investment at some point in the future Investing for single period: FV. Future Value PV. Present Value

Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt $100 At

TIME VALUE OF MONEY Return of vs. Return on Investment: We EXPECT to get more than we invest! Invest $1,000 it becomes $1,050 $1,000 return of $50 return on Factors to consider when assessing Return on

126 Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly,

TIME VALUE OF MONEY (TVM) INTEREST Rate of Return When we know the Present Value (amount today), Future Value (amount to which the investment will grow), and Number of Periods, we can calculate the rate

Discounted Cash Flow Valuation Chapter 5 Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute

Exercise 1 for Time Value of Money MULTIPLE CHOICE 1. Which of the following statements is CORRECT? a. A time line is not meaningful unless all cash flows occur annually. b. Time lines are useful for visualizing

10. Time Value of Money 2: Inflation, Real Returns, Annuities, and Amortized Loans Introduction This chapter continues the discussion on the time value of money. In this chapter, you will learn how inflation

Present Value and Annuities Chapter 3 Cont d Present Value Helps us answer the question: What s the value in today s dollars of a sum of money to be received in the future? It lets us strip away the effects

Key Concepts and Skills Chapter 4 Introduction to Valuation: The Time Value of Money Be able to compute the future value of an investment made today Be able to compute the present value of cash to be received

The Time Value of Money Time Value Terminology 0 1 2 3 4 PV FV Future value (FV) is the amount an investment is worth after one or more periods. Present value (PV) is the current value of one or more future

Investment Planning Problems on Time value of money January 22, 2015 Vandana Srivastava SENSEX closing value on Tuesday: closing value on Wednesday: opening value on Thursday: Top news of any financial

Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive $5,000 per month in retirement.

Homework 5 Solutions Chapter 4C Investment Plans. Use the savings plan formula to answer the following questions. 30. You put $200 per month in an investment plan that pays an APR of 4.5%. How much money

Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have

Copyright 2008 by the Foundation of the American College of Healthcare Executives 6/11/07 Version 9-1 CHAPTER 9 Time Value Analysis Future and present values Lump sums Annuities Uneven cash flow streams

Solutions to Time value of money practice problems Prepared by Pamela Peterson Drake 1. What is the balance in an account at the end of 10 years if $2,500 is deposited today and the account earns 4% interest,

Chapter 2 Applying Time Value Concepts Chapter Overview Albert Einstein, the renowned physicist whose theories of relativity formed the theoretical base for the utilization of atomic energy, called the

This is Time Value of Money: Multiple Flows, chapter 7 from the book Finance for Managers (index.html) (v. 0.1). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: $5,000.08 = $400 So after 10 years you will have: $400 10 = $4,000 in interest. The total balance will be

4 Annuities and Loans 4.1 Introduction In previous section, we discussed different methods for crediting interest, and we claimed that compound interest is the correct way to credit interest. This section

Applying Time Value Concepts C H A P T E R 3 based on the value of two packs of cigarettes per day and a modest rate of return? Let s assume that Lou will save an amount equivalent to the cost of two packs

Section 5.1 - Compound Interest Simple Interest Formulas If I denotes the interest on a principal P (in dollars) at an interest rate of r (as a decimal) per year for t years, then we have: Interest: Accumulated

Excel has several built in functions for working with compound interest and annuities. To use these functions, we ll start with a standard Excel worksheet. This worksheet contains the variables used throughout

NOTE: All of the information contained in this file has been collected from the various HELP files found in Excel for each of these functions. PV Returns the present value of an investment. The present

5 More on Annuities and Loans 5.1 Introduction This section introduces Annuities. Much of the mathematics of annuities is similar to that of loans. Indeed, we will see that a loan and an annuity are just

In following we will introduce one of the most important and powerful concepts you will learn in your study of finance; the time value of money. It is generally acknowledged that money has a time value.

Problems 161 The correct discount rate for a cash flow is the expected return available in the market on other investments of comparable risk and term. If the interest on an investment is taxed at rate

An annuity is a series of payments or withdrawals. ANNUITIES An Annuity can be either Simple or General Simple Annuities - Compounding periods and payment periods coincide. General Annuities - Compounding

CALCULATOR TUTORIAL INTRODUCTION Because most students that use Understanding Healthcare Financial Management will be conducting time value analyses on spreadsheets, most of the text discussion focuses

BUAD 100x Foundations of Finance Discounted Cash Flow Valuation September 28, 2009 Review Introduction to corporate finance What is corporate finance? What is a corporation? What decision do managers make?

Sample problems from Chapter 10.1 This is the annuities sinking funds formula. This formula is used in most cases for annuities. The payments for this formula are made at the end of a period. Your book

PowerPoint to accompany Chapter 5 Interest Rates 5.1 Interest Rate Quotes and Adjustments To understand interest rates, it s important to think of interest rates as a price the price of using money. When

6 Calculators Discounted Cash Flow Valuation Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute

The Anderson School at UCLA POL 2000-09 Numbers 101: Cost and Value Over Time Copyright 2000 by Richard P. Rumelt. We use the tool called discounting to compare money amounts received or paid at different

1 Excel and Mathematics of Finance Index Numbers ja Consumer Price Index The consumer Price index measures differences in the price of goods and services and calculates a change for a fixed basket of goods

Part 9. The Basics of Corporate Finance The essence of business is to raise money from investors to fund projects that will return more money to the investors. To do this, there are three financial questions

Lecture: III 1 What is a bond? Bond Valuation When a corporation wishes to borrow money from the public on a long-term basis, it usually does so by issuing or selling debt securities called bonds. A bond

D. Dimov Most financial decisions involve costs and benefits that are spread out over time Time value of money allows comparison of cash flows from different periods Question: You have to choose one of

1.-1.3 ime Value of Money and Discounted ash Flows ime Value of Money (VM) - the Intuition A cash flow today is worth more than a cash flow in the future since: Individuals prefer present consumption to

196 Part Interest Rates and Valuing Cash Flows Chapter 6 APPENDIX B The Yield Curve and the Law of One Price Thus far, we have focused on the relationship between the price of an individual bond and its